Mathematics Department
Undergraduate Research Symposium
Spring 2018

Thursday, April 26 2:45 to 4:30 pm in LCB 215


2:45-3:00 Jack Garzella
Mentor: Fernando Guevara-Vasquez
Using Spring and Mass Networks to Model an Elastic Body

2:00- 3:15 Han Le
Mentor: Tom Alberts
Spiked sample covariance matrices

3:15-3:30 Hannah Choi
Mentor: Sean Lawley
Modeling predator-prey dynamics using mean first passage time analysis

3:30-3:45 Katelyn Queen
Mentor: Fred Adler, Jason Griffiths
Dynamic Modelling of Subclones in Patients with Late-Stage Breast and Ovarian Cancers

3:45-4:00 Hannah Waddel
Mentor: Fred Adler
The Community Ecology of the Music Canon

4:00-4:15 Charlotte Blake
Mentor: Andrej Cherkaev
Approximation of Bistable Spring Chain Dynamics

4:15-4:30 Delaney Mosier
Mentor: Ken Golden
Poisson Equation Model for Sea Ice Concentration Fields in a Changing Climate


Abstracts


Charlotte Blake
Mentor: Andrej Cherkaev
Approximation of Bistable Spring Chain Dynamics

In this presentation, we address the problem of approximating the dynamics of a chain of bistable springs with one monotonic nonlinear spring. We assume piecewise linearity of the bistable springs, and identical springs in the chain separated by small masses. From there, we investigate the possible modes of behavior of the system and determine the energy bounds for each mode. We then determine the asymptotic approximations and examine the approximation with a two-spring system. Throughout the presentation, we reference numerical simulations of chains. Hannah Choi
Mentor: Sean Lawley
Modeling predator-prey dynamics using mean first passage time analysis

Animal movement structures interactions between individuals, the environment and other species, and therefore determines resource consumption, reproductive output, place of shelter, and survival of an individual. Most mathematical models seeking to understand animal movement focus on the distribution of animals and resources within a landscape. We seek to better understand movement through modeling the behavior of a single animal. We assume that animal pathways exhibit diffusive behavior and calculate the mean first passage time (MFPT) for an animal, a predator, to find a prey under various assumptions and the so-called intermittent search strategy.



Jack Garzella
Mentor: Fernando Guevara-Vasquez
Using Spring and Mass Networks to Model an Elastic Body

Given a network of masses and springs, we can easily find out how those springs will interact when nodes are displaced. However, given just the interactions of nodes on the boundary of the network, can we find out the values of all the spring constants? It turns out, in certain situations we can, using an iterative algorithm. Moreover, we can small recoverable networks to approximate bigger networks, and even a continuous elastic body.

Han Le
Mentor: Tom Alberts
Spiked sample covariance matrices

I will review the statistical theory of sample covariance matrices for random vectors, focusing on the case where the size of the sample is comparable to the dimension of the vectors. This is the situation increasingly encountered in big data. Spiked models are a way of analyzing if one component of the vector has a variance that is much larger than the others in this regime.

Delaney Mosier
Mentor: Ken Golden
Poisson Equation Model for Sea Ice Concentration Fields in a Changing Climate

The Arctic and Antarctic sea ice packs are critical regions for examining the effect and implications of a rising global temperature. Melting polar sea ice produces ecological dilemmas, an increased absorption of solar radiation, and rising sea levels that threaten coastlines. Due to their significance to the evolution of our climate system, Earth’s sea ice packs must be accurately represented in global climate models. Our study aims to enhance the representation of sea ice by examining the evolution and behavior of the sea ice concentration field as a function of both time and space. We will develop a mathematical model for the concentration field based on the Poisson equation. We will begin by exploring efficient numerical methods for solving the forward problem in one and two spatial dimensions. Our investigation of novel low-order models of the concentration field and its evolution will give us tools to mathematically analyze the changes in sea ice concentration under global warming. Our study will also begin to encompass the inverse problem for the Poisson equation, utilizing satellite data on the sea ice concentration field from 1979 to present. Given the discrete form of the concentration field on a two-dimensional lattice, we will invert for the field (the right hand side) representing the distribution of sources and sinks of ice concentration. In the analogous electrostatic problem, data on the electric potential is inverted to identify regions of positive and negative electrical charge. Our source-sink field will similarly represent sources and sinks in ice concentration - regions in which ice is primarily either freezing and converging or melting and diverging, respectively. We will explore the evolving structure of this field as the global climate has warmed and analyze the patterns we discover in shifting “hot” and “cold” spots, as well as in overall or homogenized behavior.

Katelyn Queen
Mentor: Fred Adler, Jason Griffiths
Dynamic Modelling of Subclones in Patients with Late-Stage Breast and Ovarian Cancers

During the course of chemotherapy treatment in cancer patients, genetically related groups of cells in the tumors called subclones can become resistant to treatments. This study uses mathematical and statistical models to understand this process, and eventually, we hope, to find ways to delay or even reverse the development of the chemo-resistant tumors associated with breast and ovarian cancers. Our data come from metastatic tumor cells from the pleura of patients with breast and ovarian cancer collected before, during, and after different types of treatments. These cells are then genetically sequenced with a method called single-cell RNA seq that gives the genetic composition in detail. Our current choice of mathematical tool is ordinary differential equations (ODE) models, which we are using to watch phenotypic changes during cancer growth which lead to resistance of tumors. These models will allow us to derive, simulate and analyze a dynamic model of subclone changes during the transformation of a chemo-sensitive tumor to a chemo-resistant tumor, which in turn would create the opportunity to then block or reverse this transformation in patients with late-stage breast and ovarian cancers. As an added complexity, the progression of cancers can be facilitated by non-cancerous cells like white blood cells. We will extend our data analysis techniques to determine how cancer subclones interact with other cell types, and how these interactions might be used to delay the transition to a chemo-resistant state.

Hannah Waddel
Mentor: Fred Adler
The Community Ecology of the Music Canon

Symphony orchestras today have access to a musical canon stretching back at least four hundred years and play a critical role in its establishment. Full symphony orchestras are expensive to operate and have limited performance time, which only allows a finite number of songs and composers to be considered canonical, and more music exists than time to perform it. Community ecology describes the structure and interactions of species competing for limited resources. The structure of the musical canon and its dissemination lends itself to analysis using ecological methods and models. Treating composers or songs as species, we are using quantitative ecological methods to characterize the ways that music interacts in the canon, with a particular focus on the ways that composers and pieces enter and remain in the canon. The canon follows some common ecological patterns, where a small number of species constitute the bulk of the biomass in an ecosystem and most species are rare. In the canon, a few "warhorse" composers dominate the repertoire while most new composers barely receive performance time. Our project provides a cross-disciplinary analysis of the western canon of music using mathematical and ecological methods. The efficacy of ecological models in describing the behavior of the canon lent insights into what analogous processes occur in the field of music composition that cause a composer to either fade into obscurity or become enshrined through centuries of performance.